PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations Alexandre Clément Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France [email protected] Simon Perdrix Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France [email protected] Abstract We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. Coherent control, and in particular indefinite causal order, is known to enable multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is inspired by quantum optics, in particular the polarising beam splitter (PBS for short). We formalise the syntax and the semantics of the PBS-diagrams, and we equip the language with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we consider applications like the implementation of controlled permutations and the unrolling of loops. 2012 ACM Subject Classification Theory of computation → Quantum computation theory; Theory of computation → Axiomatic semantics; Theory of computation → Categorical semantics; Hardware → Quantum computation; Hardware → Quantum communication and cryptography Keywords and phrases Quantum Computing, Diagrammatic Language, Completeness, Quantum Control, Polarising Beam Splitter, Categorical Quantum Mechanics, Quantum Switch. Related Version This is the full version of the paper published at MFCS’20 [9]. Funding This work is funded by ANR-17-CE25-0009 SoftQPro, ANR-17-CE24-0035 VanQuTe, PIA-GDN/Quantex, and LUE/UOQ. Acknowledgements The authors want to thank Mehdi Mhalla, Emmanuel Jeandel and Titouan Carette for fruitful discussions. All diagrams were written with the help of TikZit. 1 Introduction Quantum computers can solve problems which are out of reach of classical computers [29, 20]. One of the resources offered by quantum mechanics to speed up algorithms is the superposition phenomenon which allows a quantum memory to be in several possible classical states at the same time, in superposition. Less explored in quantum computing models, one can also consider a superposition of processes. Called coherent control or simply quantum control, arXiv:2002.09387v2 [quant-ph] 31 Aug 2020 it can be illustrated with the following example called quantum switch: the order in which two unitary evolutions U and V are applied is controlled by the state of a control qubit. In particular, if the control qubit is in superposition, then both UV and VU are applied, in superposition. Coherent control is loosely represented in usual formalisms of quantum computing. For instance, in the quantum circuit model, the only available quantum control is the controlled gate mechanism: a gate U is applied or not depending on the state of a control qubit. The quantum switch cannot be implemented with a single copy of U and a single copy of V in the quantum circuit model, and more generally using any language with a fixed or classically 2 PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations |↑i |→i U QS[U, V ] := V Figure 1 (a) Intuitive behaviour of a polarising beam splitter: vertical polarisation goes through, horizontal polarisation is reflected; (b) Quantum switch of two matrices U and V . controlled order of operations. Quantum switch has however been realised experimentally [25, 26]. Moreover, such a quantum control has been proved to enable various computational and communication advantages over classically ordered models [3, 15, 16, 18, 1], for instance for deciding whether two unitary transformations are commuting or anti-commuting [8] (see Example 12). Notice that other models of quantum computations (e.g. Quantum Turing Machines) or programming languages (e.g. Lineal [13] or QML [2]), allow for arbitrary coherent control of quantum evolutions, the price to pay is, however, the presence of non-trivial well-formedness conditions to ensure that the represented evolution is valid. Indeed, the superposition (i.e. linear combination) of two unitary evolutions is not necessarily a unitary evolution. We introduce a graphical language, the PBS-calculus, for representing coherent control of quantum computations, where arbitrary gates can be coherently controlled. Our goal is to provide the foundations of a formal framework which will be further developed to explore the power and limits of the coherent control of quantum evolutions. Contrary to the quantum circuit model, the PBS-calculus allows a representation of the quantum switch with a single copy of each gate to be controlled. Moreover, any PBS-diagram is valid by construction (no side nor well-formedness condition). The syntax of the PBS-diagrams is inspired by quantum optics and is actually already used in several papers dealing with coherent control of quantum evolutions [1, 3]. Our contribution is to provide formal syntax and semantics (both operational and denotational) for these diagrams, and also to introduce an equational theory which allows one to transform diagrams. Our main technical contribution is the proof that the equational theory is complete (if two diagrams have the same semantics then one can be transformed into the other using the equational theory) and minimal (in the sense that each of the equations is necessary for the completeness of the language). The syntax of the PBS-calculus is inspired by linear optics, and in particular by the peculiar behaviour of the polarising beam splitter. A polarising beam splitter transforms a superposition of polarisations into a superposition of positions: if the polarisation is vertical the photon is transmitted whereas it is reflected when the polarisation is horizontal (see Figure 1.a). As a consequence a photon can be routed in different parts of a scheme, this routing being quantumly controlled by the polarisation of the photon. This is a unique behaviour which has no counterpart in the quantum circuit model for instance. Polarising beam splitters can be used to perform a quantum switch, as depicted as a PBS-diagram in Figure 1.b. Related works. In the context of categorical quantum mechanics several graphical languages have already been introduced: ZX-calculus [10, 21], ZW-calculus [19], ZH-calculus [4] and their variants. Notice in particular a proposal for representing fermionic (non polarising) beam splitters in the ZW-calculus [12]. An apparent difference between the PBS-calculus and these languages, is that the category of PBS-diagrams is traced but not compact closed. This difference is probably not fundamental, as for any traced monoidal category there is a completion of it to a compact closed category [22]. The fundamental difference is the parallel composition: in the PBS-calculus two parallel wires correspond to two possible positions of a A. Clément and S. Perdrix 3 single particle (i.e. a direct sum in terms of semantics), whereas, in the other languages it corresponds to two particles (i.e. a tensor product). The parallel composition makes the PBS-calculus closer to the graphical linear algebra approach [7, 6, 5], however the generators and the fundamental structures (e.g. Frobenius algebra, Hopf algebra) are a priori unrelated to those of the PBS-calculus. In the context of quantum programming languages, there are a few proposals for represent- ing quantum control [13, 2, 31, 27]. Colnaghi et al. [11] have introduced a graphical language with programmable connections. The language uses the quantum switch as a generator, but does not aim to describe schemes with polarising beam splitters. Notice also that the inputs/outputs of the language are quantum channels. Structure of the paper. In Section 2, the syntax of the PBS-diagrams is introduced. The PBS-diagrams are considered up to a structural congruence which allows one to deform the diagrams at will. Section 3 is dedicated to the semantics of the language: two semantics, a path semantics and a denotational semantics, are introduced. The denotational semantics is proved to be adequate with respect to the path semantics. In Section 4, the axiomatisation of the PBS-calculus is introduced, and our main result, the soundness and completeness of the language, is proved. The axiomatisation is also proved to be minimal in the sense that none of the axioms can be derived from the others. Finally, in Section 5, we consider the application of the PBS-calculus to the problem of loop unrolling. We show in particular that any PBS-diagram involving unitary matrices can be transformed into a trace-free diagram. The paper is written such that the reader does not need any particular knowledge in category theory. Basic definitions, in particular of Traced PROP, are however given in Appendix A for completeness. 2 Syntax A PBS-diagram is made of polarising beam splitters , polarisation flips ¬ , and q×q gates U for any matrix U ∈ C , where q is a fixed positive integer. One can also use wires like the identity or the swap . Diagrams can be combined by means of sequential composition ◦, parallel composition ⊗, and trace T r(·). The trace consists in connecting the last output of a diagram to its last input, like a feedback loop. The symbol represents the empty diagram. Any diagram has a type n → n which corresponds to the numbers of input/output wires. The syntax of the language is the following: I Definition 1. Given q ∈ N \{0}, a PBSq-diagram D : n → n is inductively defined as: : 0 → 0 : 1 → 1 ¬ : 1 → 1 : 2 → 2 : 2 → 2 q×q U ∈ C D1 : n→n D2 : n→n D1 : n→n D2 : m→m D : n+1 → n+1 U : 1→1 D2 ◦ D1 : n→n D1 ⊗ D2 : n+m → n+m T r(D): n → n Sequential composition D2 ◦ D1, parallel composition D1 ⊗ D2, and trace T r(D) are respect- ively depicted as follows: · · · D1 · · · · · · · D1 · D2 · · D · · · · D2 · In the following, the positive integer q will be omitted when it is useless or clear from the context.